#6826676

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1256 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,29)(20,33)(21,36)(22,26)(23,28)(24,31)(25,32)(27,38)(30,40)(34,41)(35,42)(37,43)(39,48)(44,47)(45,46);(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,26,29)(20,34,32,23)(21,35,38,37)(25,41,33,28)(27,42,36,43)(30,39,46,47)(40,44,45,48);(2,5)(6,8)(11,16)(13,15)(17,20,23,19,22,32,34,31)(18,25,28,24,26,33,41,29)(21,37,39,46,38,35,47,30)(27,43,44,45,36,42,48,40);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,26)(21,38)(24,29)(25,33)(28,41)(30,46)(35,37)(39,47);(17,21,23,39,22,38,34,47)(18,27,28,44,26,36,41,48)(19,30,32,37,31,46,20,35)(24,40,33,43,29,45,25,42)

Polycyclic presentation:

2^9|b^2=i,c^2=fhi,d^2=gi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6733872);